# §36.2 Catastrophes and Canonical Integrals

## §36.2(i) Definitions

### Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$

 36.2.1 $\mathop{\Phi_{K}\/}\nolimits\!\left(t;\mathbf{x}\right)=t^{K+2}+\sum_{m=1}^{K}% x_{m}t^{m}.$ Defines: $\mathop{\Phi_{\NVar{K}}\/}\nolimits\!\left(\NVar{t};\NVar{\mathbf{x}}\right)$: cuspoid catastrophe of codimension $K$ Symbols: $n$: integer, $t$: variable, $K$: codimension and $x_{i}$: real parameter Referenced by: §36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E1 Encodings: TeX, pMML, png See also: Annotations for 36.2(i)

Special cases: $K=1$, fold catastrophe; $K=2$, cusp catastrophe; $K=3$, swallowtail catastrophe.

### Normal Forms for Umbilic Catastrophes with Codimension $K=3$

 36.2.2 $\displaystyle\mathop{\Phi^{(\mathrm{E})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$ $\displaystyle=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt+xs,$ $\mathbf{x}=\{x,y,z\}$, Defines: $\mathop{\Phi^{(\mathrm{E})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$: elliptic umbilic catastrophe Symbols: $y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter Referenced by: §36.10(iii), §36.2(i), §36.7(iii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E2 Encodings: TeX, pMML, png See also: Annotations for 36.2(i) (elliptic umbilic). 36.2.3 $\displaystyle\mathop{\Phi^{(\mathrm{H})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$ $\displaystyle=s^{3}+t^{3}+zst+yt+xs,$ $\mathbf{x}=\{x,y,z\}$, Defines: $\mathop{\Phi^{(\mathrm{H})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$: hyperbolic umbilic catastrophe Symbols: $y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter Referenced by: §36.10(iii), §36.2(i), §36.8 Permalink: http://dlmf.nist.gov/36.2.E3 Encodings: TeX, pMML, png See also: Annotations for 36.2(i) (hyperbolic umbilic).

### Canonical Integrals

 36.2.4 $\mathop{\Psi_{K}\/}\nolimits\!\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}% \mathop{\exp\/}\nolimits\!\left(i\mathop{\Phi_{K}\/}\nolimits\!\left(t;\mathbf% {x}\right)\right)\mathrm{d}t.$ Defines: $\mathop{\Psi_{\NVar{K}}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: canonical integral function Symbols: $\mathop{\Phi_{\NVar{K}}\/}\nolimits\!\left(\NVar{t};\NVar{\mathbf{x}}\right)$: cuspoid catastrophe of codimension $K$, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\int$: integral, $t$: variable and $K$: codimension Referenced by: §36.12(i), §36.2(i), §36.7(ii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E4 Encodings: TeX, pMML, png See also: Annotations for 36.2(i)
 36.2.5 $\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!\left(\mathbf{x}\right)=\int_{-\infty% }^{\infty}\int_{-\infty}^{\infty}\mathop{\exp\/}\nolimits\!\left(i\mathop{\Phi% ^{(\mathrm{U})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)\right)\mathrm{d}s% \mathrm{d}t,$ $\mathrm{U}=\mathrm{E},\mathrm{H}$. Defines: $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: elliptic umbilic canonical integral function, $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: hyperbolic umbilic canonical integral function and $\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: umbilic canonical integral function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\int$: integral, $\mathop{\Phi^{(\mathrm{U})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$, $t$: variable and $s$: variable Referenced by: §36.10(iii), §36.10(iv), §36.2(i), §36.7(iii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E5 Encodings: TeX, pMML, png See also: Annotations for 36.2(i)
 36.2.6 $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\mathbf{x}\right)=2\sqrt{\ifrac% {\pi}{3}}\,\mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{4}{27}z^{3}+\tfrac{1}% {3}xz-\tfrac{1}{4}\pi\right)\right)\int_{\infty\mathop{\exp\/}\nolimits\!\left% (-7\pi i/12\right)}^{\infty\mathop{\exp\/}\nolimits\!\left(\pi i/12\right)}% \mathop{\exp\/}\nolimits\!\left(i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2% }}{12u^{2}}\right)\right)\mathrm{d}u,$

with the contour passing to the lower right of $u=0$.

 36.2.7 $\displaystyle\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\mathbf{x}\right)$ $\displaystyle=\dfrac{4\pi}{3^{1/3}}\mathop{\exp\/}\nolimits\!\left(i\left(% \tfrac{2}{27}z^{3}-\tfrac{1}{3}xz\right)\right)\left(\mathop{\exp\/}\nolimits% \!\left(-i\dfrac{\pi}{6}\right)\mathrm{F}_{+}(\mathbf{x})+\mathop{\exp\/}% \nolimits\!\left(i\dfrac{\pi}{6}\right)\mathrm{F}_{-}(\mathbf{x})\right),$ $\displaystyle\mathrm{F}_{\pm}(\mathbf{x})$ $\displaystyle=\int_{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(ry\mathop{\exp% \/}\nolimits\!\left(\pm i\dfrac{\pi}{6}\right)\right)\mathop{\exp\/}\nolimits% \!\left(2ir^{2}z\mathop{\exp\/}\nolimits\!\left(\pm i\dfrac{\pi}{3}\right)% \right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(3^{2/3}r^{2}+3^{-1/3}\mathop{% \exp\/}\nolimits\!\left(\mp i\dfrac{\pi}{3}\right)\left(\tfrac{1}{3}z^{2}-x% \right)\right)\mathrm{d}r.$
 36.2.8 $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(\mathbf{x}\right)=4\sqrt{\ifrac% {\pi}{6}}\,\mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{1}{27}z^{3}+\tfrac{1}% {6}z(y+x)+\tfrac{1}{4}\pi\right)\right)\*\int_{\infty\mathop{\exp\/}\nolimits% \!\left(5\pi i/12\right)}^{\infty\mathop{\exp\/}\nolimits\!\left(\pi i/12% \right)}\mathop{\exp\/}\nolimits\!\left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{% 2}z^{2}+x+y\right)u^{2}-\frac{(y-x)^{2}}{24u^{2}}\right)\right)\mathrm{d}u,$

with the contour passing to the upper right of $u=0$.

 36.2.9 $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(\mathbf{x}\right)=\frac{2\pi}{3% ^{1/3}}\int_{\infty\mathop{\exp\/}\nolimits\!\left(5\pi i/6\right)}^{\infty% \mathop{\exp\/}\nolimits\!\left(\pi i/6\right)}\mathop{\exp\/}\nolimits\!\left% (i(s^{3}+xs)\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{zs+y}{3^{1/3}}% \right)\mathrm{d}s.$

### Diffraction Catastrophes

 36.2.10 $\mathop{\Psi_{K}\/}\nolimits\!(\mathbf{x};k)=\sqrt{k}\int_{-\infty}^{\infty}% \mathop{\exp\/}\nolimits\!\left(ik\mathop{\Phi_{K}\/}\nolimits\!\left(t;% \mathbf{x}\right)\right)\mathrm{d}t,$ $k>0$. Defines: $\mathop{\Psi_{\NVar{K}}\/}\nolimits\!(\NVar{\mathbf{x}};k)$: diffraction catastrophe Symbols: $\mathop{\Phi_{\NVar{K}}\/}\nolimits\!\left(\NVar{t};\NVar{\mathbf{x}}\right)$: cuspoid catastrophe of codimension $K$, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\int$: integral, $k$: variable, $t$: variable and $K$: codimension Referenced by: §36.12(i) Permalink: http://dlmf.nist.gov/36.2.E10 Encodings: TeX, pMML, png See also: Annotations for 36.2(i)
 36.2.11 $\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!(\mathbf{x};k)=k\int_{-\infty}^{% \infty}\int_{-\infty}^{\infty}\mathop{\exp\/}\nolimits\!\left(ik\mathop{\Phi^{% (\mathrm{U})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)\right)\mathrm{d}s% \mathrm{d}t,$ $\mathrm{U=E,H}$; $k>0$. Defines: $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!(\NVar{\mathbf{x}};\NVar{k})$: elliptic umbilic canonical integral function, $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!(\NVar{\mathbf{x}};\NVar{k})$: hyperbolic umbilic canonical integral function and $\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!(\NVar{\mathbf{x}};\NVar{k})$: umbilic canonical integral function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\int$: integral, $\mathop{\Phi^{(\mathrm{U})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$, $k$: variable, $t$: variable and $s$: variable Referenced by: §36.2(i) Permalink: http://dlmf.nist.gov/36.2.E11 Encodings: TeX, pMML, png See also: Annotations for 36.2(i)

For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975).

## §36.2(ii) Special Cases

 36.2.12 $\mathop{\Psi_{0}\/}\nolimits=\sqrt{\pi}\mathop{\exp\/}\nolimits\!\left(i\frac{% \pi}{4}\right).$

$\mathop{\Psi_{1}\/}\nolimits$ is related to the Airy function (§9.2):

 36.2.13 $\mathop{\Psi_{1}\/}\nolimits\!\left(x\right)=\frac{2\pi}{3^{1/3}}\mathop{% \mathrm{Ai}\/}\nolimits\!\left(\frac{x}{3^{1/3}}\right).$

$\mathop{\Psi_{2}\/}\nolimits$ is the Pearcey integral (Pearcey (1946)):

 36.2.14 $\mathop{\Psi_{2}\/}\nolimits\!\left(\mathbf{x}\right)=P(x_{2},x_{1})=\int_{-% \infty}^{\infty}\mathop{\exp\/}\nolimits\!\left(\mathrm{i}(t^{4}+x_{2}t^{2}+x_% {1}t)\right)\mathrm{d}t.$

(Other notations also appear in the literature.)

 36.2.15 $\mathop{\Psi_{K}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{K+2}\right)\*\begin{cases}\mathop{% \exp\/}\nolimits\!\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\ \mathop{\cos\/}\nolimits\!\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}$
 36.2.16 $\displaystyle\mathop{\Psi_{1}\/}\nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=1.54669,$ $\displaystyle\mathop{\Psi_{2}\/}\nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=1.67481+\mathrm{i}\,0.69373$ $\displaystyle\mathop{\Psi_{3}\/}\nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=1.74646,$ $\displaystyle\mathop{\Psi_{4}\/}\nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=1.79222+\mathrm{i}\,0.48022.$ Symbols: $\mathop{\Psi_{\NVar{K}}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: canonical integral function Permalink: http://dlmf.nist.gov/36.2.E16 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 36.2(ii)
 36.2.17 $\displaystyle\frac{{\partial}^{p}}{{\partial x_{1}}^{p}}\mathop{\Psi_{K}\/}% \nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=\frac{2}{K+2}\mathop{\Gamma\/}\nolimits\!\left(\frac{p+1}{K+2}% \right)\mathop{\cos\/}\nolimits\!\left(\frac{\pi}{2}\left(\frac{p+1}{K+2}+p% \right)\right),$ $K$ odd, $\displaystyle\frac{{\partial}^{2q+1}}{{\partial x_{1}}^{2q+1}}\mathop{\Psi_{K}% \/}\nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=0,$ $K$ even, $\displaystyle\frac{{\partial}^{2q}}{{\partial x_{1}}^{2q}}\mathop{\Psi_{K}\/}% \nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=\frac{2}{K+2}\mathop{\Gamma\/}\nolimits\!\left(\frac{2q+1}{K+2}% \right)\mathop{\exp\/}\nolimits\!\left(i\frac{\pi}{2}\left(\frac{2q+1}{K+2}+2q% \right)\right),$ $K$ even.
 36.2.18 $\displaystyle\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\boldsymbol{{0}}\right)$ $\displaystyle=\tfrac{1}{3}\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1% }{6}\right)=3.28868,$ $\displaystyle\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(0\right)$ $\displaystyle=\tfrac{1}{3}{\mathop{\Gamma\/}\nolimits^{2}}\!\left(\tfrac{1}{3}% \right)=2.39224.$
 36.2.19 $\mathop{\Psi_{2}\/}\nolimits\!\left(0,y\right)=\frac{\pi}{2}\sqrt{\frac{|y|}{2% }}\mathop{\exp\/}\nolimits\!\left(-i\frac{y^{2}}{8}\right)\left(\mathop{\exp\/% }\nolimits\!\left(i\frac{\pi}{8}\right)\mathop{J_{-\ifrac{1}{4}}\/}\nolimits\!% \left(\frac{y^{2}}{8}\right)-\mathop{\mathrm{sign}\/}\nolimits\!\left(y\right)% \mathop{\exp\/}\nolimits\!\left(-i\frac{\pi}{8}\right)\mathop{J_{\ifrac{1}{4}}% \/}\nolimits\!\left(\frac{y^{2}}{8}\right)\right).$

For the Bessel function $\mathop{J\/}\nolimits$ see §10.2(ii).

 36.2.20 $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(x,y,0\right)=2\pi^{2}(\tfrac{2}% {3})^{2/3}\Re{\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{x+iy}{12^{1/3% }}\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(\frac{x-iy}{12^{1/3}}\right)% \right)},$
 36.2.21 $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(x,y,0\right)=\frac{4\pi^{2}}{3^% {2/3}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{x}{3^{1/3}}\right)\mathop{% \mathrm{Ai}\/}\nolimits\!\left(\frac{y}{3^{1/3}}\right).$

Addendum: For further special cases see §36.2(iv)

## §36.2(iii) Symmetries

 36.2.22 $\mathop{\Psi_{2K}\/}\nolimits\!\left(\mathbf{x}^{\prime}\right)=\mathop{\Psi_{% 2K}\/}\nolimits\!\left(\mathbf{x}\right),$ $x_{2m+1}^{\prime}=-x_{2m+1}$, $x_{2m}^{\prime}=x_{2m}$.
 36.2.23 $\mathop{\Psi_{2K+1}\/}\nolimits\!\left(\mathbf{x}^{\prime}\right)={\mathop{% \Psi_{2K+1}\/}\nolimits^{\ast}}\!\left(\mathbf{x}\right),$ $x_{2m+1}^{\prime}=x_{2m+1}$, $x_{2m}^{\prime}=-x_{2m}$.
 36.2.24 $\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!\left(x,y,z\right)={\Psi^{\ast}}^{(% \mathrm{U})}(x,y,-z),$ $\mathrm{U=E,H}$.
 36.2.25 $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(x,-y,z\right)=\mathop{\Psi^{(% \mathrm{E})}\/}\nolimits\!\left(x,y,z\right).$
 36.2.26 $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3% }}{2}y,\pm\tfrac{\sqrt{3}}{2}x-\tfrac{1}{2}y,z\right)=\mathop{\Psi^{(\mathrm{E% })}\/}\nolimits\!\left(x,y,z\right),$

(rotation by $\pm\tfrac{2}{3}\pi$ in $x,y$ plane).

 36.2.27 $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(x,y,z\right)=\mathop{\Psi^{(% \mathrm{H})}\/}\nolimits\!\left(y,x,z\right).$

## §36.2(iv) Addendum to 36.2(ii) Special Cases

 36.2.28 $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(0,0,z\right)={\Psi^{\ast}}^{(% \mathrm{E})}(0,0,-z)\\ =2\pi\sqrt{\frac{\pi z}{27}}\mathop{\exp\/}\nolimits\!\left(\frac{2}{27}iz^{3}% \right)\*\left(\mathop{J_{-1/6}\/}\nolimits\!\left(\frac{2}{27}z^{3}\right)+i% \mathop{J_{1/6}\/}\nolimits\!\left(\frac{2}{27}z^{3}\right)\right),$ $z\geq 0$, Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: elliptic umbilic canonical integral function, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function and $z$: real parameter Referenced by: Other Changes Permalink: http://dlmf.nist.gov/36.2.E28 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation has been added. For the proof see Berry and Howls (2010). Reported 2012-04-02 See also: Annotations for 36.2(iv)
 36.2.29 $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(0,0,z\right)={\Psi^{\ast}}^{(% \mathrm{H})}(0,0,-z)=\frac{2^{1/3}}{\sqrt{3}}\mathop{\exp\/}\nolimits\!\left(% \frac{1}{27}iz^{3}\right)\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(0,0,-% \frac{z}{2^{2/3}}\right),$ $-\infty. Symbols: $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: elliptic umbilic canonical integral function, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$: hyperbolic umbilic canonical integral function and $z$: real parameter Referenced by: Other Changes Permalink: http://dlmf.nist.gov/36.2.E29 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation has been added. For the proof see Berry and Howls (2010). Reported 2012-05-01 See also: Annotations for 36.2(iv)

Here the functions ${\Psi^{\ast}}^{(\mathrm{E})}$ and ${\Psi^{\ast}}^{(\mathrm{H})}$ are the complex conjugates of the functions $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits$ and $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits$, respectively.